Nonlinear Partial Differential Equations

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Former 2008/2009 Natural Sciences - Medicine - Mathematics

Abstract

The research program will focus on the mathematical discipline nonlinear partial differential equation. Differential equations have their origin in the quest to describe nature by mathematics, and these equations are perfect tools to describe physical phenomena that vary in space and time. In fact, all the fundamental laws of nature are given by differential equations (Newton’s law describes gravitation, Maxwell’s equations describe electromagnetism, Navier–Stokes’ equations describe fluid flow, etc). Thus they model a wide variety of phenomena, and have been extensively studied by mathematicians, physicists, engineers, and others. Differential equations constitute a core mathematical discipline, cf. the Abel Prize 2005. However, several key questions have not been answered, or only been partly answered. Central issues are: • Do the equations have a solution? It is still not known if the equations that describe the flow of air around an airplane have a solution. • Given that the equation has a solution, is it unique? For flow of oil in petroleum reservoirs the equations give several possible distinct values for the saturation at the same point in space. • Is the solution stable? Will a small change in one place in space have dramatic consequences in a completely different location (“the butterfly effect"). • If the equation has a unique and stable solution, how can one determine the solution? Given the measured weather data for today, how can one predict the weather for tomorrow? Variations of these questions for carefully selected nonlinear partial differential equations constitute the focus for the research program, in particular the interplay between theoretical results (mathematical properties of the solution) and numerical computations (how to compute the solution). Differential equations have been an intensive research area in mathematics in Norway the last two decades, and there are now strong research groups in Oslo, Trondheim, and Bergen. The research program will both benefit from the strong national activities and further strengthen it. Furthermore, applications to other areas of science will have impact there. The generic nature of mathematics means that the methods and techniques of this program can be applied to several seemingly distinct areas. We will focus on four applications, namely flow in porous media (e.g., flow of oil in a petroleum reservoir), mixtures of solids and fluids (e.g., sedimentation), gas dynamics (e.g., flow of gas around obstacles), water waves (e.g., breaking of waves).

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